TITLE: Ways of showing $\sum_\limits{n=1}^{\infty}\ln(1+1/n)$ to be divergent
QUESTION [6 upvotes]: Show that the following sum is divergent
  $$\sum_{n=1}^{\infty}\ln\left(1+\frac1n\right)$$

I thought to do this using Taylor series using the fact that
$$
\ln\left(1+\frac1n\right)=\frac1n+O\left(\frac1{n^2}\right)
$$
Which then makes it clear that
$$
\sum_{n=1}^{\infty}\ln\left(1+\frac1n\right)\sim \sum_{n=1}^{\infty}\frac1n\longrightarrow \infty
$$
But I feel like I overcomplicated the problem and would be interested to see some other solutions. Also, would taylor series be the way you would see that this diverges if you were not told?

REPLY [2 votes]: In THIS ANSWER, I showed using elementary analysis only that the logarithm function satisfies the inequalities 
$$\frac{x-1}{x}\le\log(x)\le x-1\tag1$$
Letting $x=1+\frac1n$ in $(1)$ reveals 
$$\frac1{n+1}\le\log\left(1+\frac1n\right)\le \frac1n\tag2$$
And using the left-hand side of $(2)$ shows that
$$\sum_{n=1}^N \log\left(1+\frac1n\right)\ge \sum_{n=1}^N \frac1{n+1}$$
Inasmuch as the harmonic series diverges, we see that the series of interest diverges also.  And we are done!